Plans and models

The plane and space, grade 11 (3 units) Lesson 2 of 15Learning

Plans and models

The plan of a flat

A flat’s plan, a drawing of a piece of furniture and a model car are all made to a scale.

On a plan at a scale of $\displaystyle 1:100$ , every centimetre on the plan is 100 cm – one metre in the real world.
On a plan at a scale of $\displaystyle 1:50$ , every centimetre is 50 cm – half a metre.
Plans are measured in centimetres and metres, not in kilometres.

Example: a room’s measurements

This is the plan of a room at a scale of $\displaystyle 1:50$ . Every square on it is 1 cm by 1 cm.

How long and how wide is the room in reality, in metres?

On the plan the room is 7 cm by 5 cm.
Every centimetre on the plan is 50 cm in reality: $\displaystyle {7\cdot50}={350}$ cm and $\displaystyle {5\cdot50}={250}$ cm.
The room is 3.5 m by 2.5 m.

Exercise 1

On a flat’s plan at a scale of $\displaystyle 1:50$ , the living room is a rectangle 11 cm by 8.4 cm.
What are the living room’s real measurements, in metres?

a

Its length

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Its width

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A sofa 2.2 m long is drawn on the same plan, at a scale of $\displaystyle 1:50$ .
How long is the sofa on the plan, in centimetres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

A model

A model car at a scale of $\displaystyle 1:18$ is 18 times smaller than the car in every direction – in length, in width and in height.

A car 4.5 m long is 450 cm long.
The model’s length: $\displaystyle {450\div18}={25}$ cm.

Exercise 3

A model car at a scale of $\displaystyle 1:43$ is 10 cm long.
How long is the real car, in metres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The scale of an enlargement

When something very small is drawn – an insect, a part of a watch, a screw – the drawing is larger than the thing, and the first number of the scale is the larger one.

At a scale of $\displaystyle 5:1$ , every centimetre in reality is 5 cm on the drawing.
A screw 8 mm long is $\displaystyle {8\cdot5}={40}$ mm long on the drawing, that is 4 cm.
Here too the ratio is a length on the drawing to the length in reality.

Exercise 4

A biology book has a drawing of an ant at a scale of $\displaystyle 10:1$ .
The ant is 6.5 cm long in the drawing.
How long is the ant in reality, in millimetres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

A bedroom 6 m by 4 m is to be drawn as large as possible on an A4 sheet – 29.7 cm by 21 cm.
Which scale fits: $\displaystyle 1:10$ , $\displaystyle 1:20$ or $\displaystyle 1:50$ ?
Explain.

Formulas: the plane and space, grade 11 (3 units)

Scale

The real length – $\displaystyle l$ the length on the map, at a scale of $\displaystyle 1:n$
$\displaystyle L$ $\displaystyle =$ $\displaystyle n\cdot l$
The real area – $\displaystyle s$ the area on the map, at a scale of $\displaystyle 1:n$
$\displaystyle S$ $\displaystyle =$ $\displaystyle n^{2}\cdot s$

Similar shapes

The scale factor – $\displaystyle a_1$ , $\displaystyle a_2$ corresponding sides
$\displaystyle k$ $\displaystyle =$ $\displaystyle \frac{a_2}{a_1}$
The ratio of the areas of similar shapes
$\displaystyle \frac{S_2}{S_1}$ $\displaystyle =$ $\displaystyle k^{2}$
A height by its shadow – $\displaystyle H$ , $\displaystyle S$ the object and its shadow; $\displaystyle h$ , $\displaystyle s$ a stick and its shadow
$\displaystyle \frac{H}{S}$ $\displaystyle =$ $\displaystyle \frac{h}{s}$

Trigonometry in the right-angled triangle

Sine – $\displaystyle a$ the side opposite the angle, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
Cosine – $\displaystyle b$ the side adjacent to the angle
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
Tangent – the opposite side over the adjacent side
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle from its tangent (and likewise $\displaystyle \sin^{-1}$ , $\displaystyle \cos^{-1}$ )
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \tan^{-1}\frac{a}{b}$
Pythagoras’ theorem
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
A gradient in percent – $\displaystyle \alpha$ the angle with the horizontal
$\displaystyle p$ $\displaystyle =$ $\displaystyle 100\cdot\tan\alpha$